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Concept of Coherence in Neutrino and Antineutrino Scattering off Nuclei

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Abstract

The concept of coherence in the scattering of neutrinos and antineutrinos off nuclei is discussed. Motivated by the results of the COHERENT experiment, a new approach to coherence in these processes is proposed, which allows a unified description of the elastic (coherent) and inelastic (incoherent) contributions to the total cross section for neutrino and antineutrino scattering off nuclei at energies below 100 MeV. Experiments and physical problems for coherent scattering of (anti)neutrinos off nuclei are briefly discussed. The extended appendix covers the main points and conclusions of the proposed approach in pedagogical detail.

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Notes

  1. In [3], the role of coherent neutrino scattering off the nuclear medium in supernova evolution is discussed. Formulas are given for the overall amplitude of coherent scattering off spin-zero (\(J = 0\)) nuclei with \(Z = N\)

    $$M({\mathbf{k}}{\kern 1pt} ',{\mathbf{k}}) \sim GA{{e}^{{ - {{q}^{2}}{{R}^{2}}}}}( - {{\sin}^{2}}{{\theta }_{W}}){{\bar {u}}_{l}}(\nu {\kern 1pt} '){{\gamma }^{0}}{{u}_{\nu }}(\nu ),\,\,\,\,\,\,\,\,\,\,\,\,$$
    (12)

    and for differential and total cross sections

    $$\begin{gathered} \frac{{d{{\sigma }_{A}}}}{{d\cos\theta }} \simeq \frac{{{{\sigma }_{0}}}}{8}{{( - {{\sin}^{2}}{{\theta }_{W}})}^{2}}{{A}^{2}}{{\omega }^{2}}(1 + \cos\theta ),\,\,\, \\ {{\sigma }_{A}} \simeq \frac{{{{\sigma }_{0}}}}{4}{{( - {{\sin}^{2}}{{\theta }_{W}})}^{2}}{{A}^{2}}{{\omega }^{2}}. \\ \end{gathered} $$

    The neutrino energy is assumed to be \(\omega \ll {{m}_{A}}\) (point nucleus). If \(J \ne 0\) and \(Z \ne N\), it is necessary to sum over individual nucleons

    $$\begin{gathered} M({\mathbf{k}}{\kern 1pt} ',{\mathbf{k}}) \sim G{{e}^{{ - {{q}^{2}}{{R}^{2}}}}}\left[ { - {{\sin}^{2}}{{\theta }_{W}}(Z + N) + \frac{{1 - 2{{\sin}^{2}}{{\theta }_{W}}}}{2}(Z - N)} \right. \\ \left. { - \frac{{{{g}_{V}}}}{2}({{Z}_{ \uparrow }} - {{Z}_{ \downarrow }}) + \frac{{{{g}_{A}}}}{2}({{N}_{ \uparrow }} - {{N}_{ \downarrow }})} \right]\bar {u}(\nu {\kern 1pt} '){{\gamma }^{0}}u(\nu ), \\ \end{gathered} $$

    where the arrow indicates nucleons with spins directed upward and downward that are scattered via axial neutral current. For all nuclei (except very light ones) the most important term in \(M({\mathbf{k}}{\kern 1pt} ',{\mathbf{k}})\) is proportional to \(( - {{\sin}^{2}}{{\theta }_{W}})(Z + N)\), in other words, relation (13) is usually valid. For spin-zero (\(J = 0\)) nuclei with \(Z \ne N\) there is a relation to the nuclear isospin \({{I}_{3}} = \tfrac{1}{2}(Z - N)\) in the form

    $$\begin{gathered} \frac{{d\sigma _{A}^{{J = 0}}}}{{d\cos\theta }} \simeq \frac{{{{\sigma }_{0}}}}{8}{{A}^{2}}si{{n}^{4}}{{\theta }_{W}}\mathop {\left[ {1 + \frac{{1 - 2{{\sin}^{2}}{{\theta }_{W}}}}{{ - {{\sin}^{2}}{{\theta }_{W}}}}\frac{{{{I}_{3}}}}{A}} \right]}\nolimits^2 {{\omega }^{2}}(1 + \cos\theta ),\,\,\, \\ \sigma _{A}^{{J = 0}} \simeq \frac{{{{\sigma }_{0}}}}{4}si{{n}^{4}}{{\theta }_{W}}{{A}^{2}}{{[1 + ...]}^{2}}{{\omega }^{2}}. \\ \end{gathered} $$

  2. Perhaps, these works are the first where attention is explicitly paid to the role of inelastic processes in coherent neutrino scattering. In [19], this role can be thought of as being formulated quantitatively.

  3. In the Standard Model, \(g_{V}^{e} = \frac{1}{2} + 2{{\sin}^{2}}{{\theta }_{W}},\,\,g_{V}^{p} = \frac{1}{2} - \) \(2{{\sin}^{2}}{{\theta }_{W}},\) \(g_{V}^{n} = - \frac{1}{2},\) \(g_{A}^{e} = \frac{1}{2},\) \(g_{A}^{p} = \frac{{{{g}_{A}}}}{2},\) \(g_{A}^{n} = - \frac{{{{g}_{A}}}}{2}.\)

  4. That is, by averaging over the initial projections of the spin and summation over all final projections of the nuclear spin, as is clearly seen in formula (23). Unfortunately, nothing is said about invariability of the final state of the nucleus, which is necessary for fulfillment of the coherence condition. This problem is discussed in more detail in appendix 8.1.

  5. In [34], there is another parameter used to characterize (partial) coherence, the relative cross section variation \(\xi \equiv \frac{{\sigma (\alpha )}}{{\sigma (\alpha = 1)}} = \alpha + (1 - \alpha )\frac{{({{\varepsilon }^{2}}Z + N)}}{{{{{(\varepsilon Z - N)}}^{2}}}}\), which linearly depends on \(\alpha \), and they both are unity in the case of complete coherence.

  6. Indeed, if in, say, the two-nucleon case there is symmetry \(\psi _{n}^{{( * )}}({\mathbf{y}},{\mathbf{x}}) = \) \(( \pm )\psi _{n}^{{( * )}}({\mathbf{x}},{\mathbf{y}})\), then \(f_{{mn}}^{{k = 1}}({\mathbf{q}}) \equiv \) \(\int {d{{{\mathbf{x}}}_{1}}} d{{{\mathbf{x}}}_{2}}\psi _{m}^{ * }({{{\mathbf{x}}}_{1}},{{{\mathbf{x}}}_{2}}){{\psi }_{n}}({{{\mathbf{x}}}_{1}},{{{\mathbf{x}}}_{2}}){{e}^{{i{\mathbf{q}}{{{\mathbf{x}}}_{1}}}}}\) can be written as \(\int {d{{{\mathbf{x}}}_{1}}} d{\mathbf{y}}\psi _{m}^{ * }({{{\mathbf{x}}}_{1}},{\mathbf{y}}){{\psi }_{n}}({{{\mathbf{x}}}_{1}},{\mathbf{y}}){{e}^{{i{\mathbf{q}}{{{\mathbf{x}}}_{1}}}}}\) and after designating \({{{\mathbf{x}}}_{1}}\) as \({{{\mathbf{x}}}_{2}}\) one arrives at \(f_{{mn}}^{{k = 1}}({\mathbf{q}})\) = \(\int {d{{{\mathbf{x}}}_{2}}d{\mathbf{y}}} \psi _{m}^{ * }({{{\mathbf{x}}}_{2}},{\mathbf{y}}){{\psi }_{n}}({{{\mathbf{x}}}_{2}},{\mathbf{y}}){{e}^{{i{\mathbf{q}}{{{\mathbf{x}}}_{2}}}}}\). Considering the abovementioned wave function symmetry condition and permutation of the integration variables, this expression takes the form \(\int {d{\mathbf{y}}d{{{\mathbf{x}}}_{2}}} \psi _{m}^{ * }({\mathbf{y}},{{{\mathbf{x}}}_{2}}){{\psi }_{n}}({\mathbf{y}},{{{\mathbf{x}}}_{2}}){{e}^{{i{\mathbf{q}}{{{\mathbf{x}}}_{2}}}}}\), which is indistinguishable from the definition of \(f_{{mn}}^{{k = 2}}({\mathbf{q}})\).

  7. \({{\left| \mathcal{A} \right|}^{2}} = \sum\nolimits_m {{{{\left| {{{\mathcal{A}}_{{mn}}}} \right|}}^{2}}} \) = \({{\left| {{{\mathcal{A}}_{0}}} \right|}^{2}}\sum\nolimits_{k,j} {\sum\nolimits_m {f_{{mn}}^{{j,\dag }}} } ({\mathbf{q}})f_{{mn}}^{k}({\mathbf{q}})\) = \({{\left| {{{\mathcal{A}}_{0}}} \right|}^{2}}\sum\nolimits_{k,j} {\left\langle n \right|} {{e}^{{ - i{\mathbf{q}}\mathop {{\mathbf{\hat {X}}}}\nolimits_j }}}\sum\nolimits_m {\left| m \right\rangle } \left\langle m \right|{{e}^{{i{\mathbf{q}}\mathop {{\mathbf{\hat {X}}}}\nolimits_k }}}\left| n \right\rangle \to (40).\)

  8. Indeed, the quantity \({{\left| \mathcal{A} \right|}^{2}}\) divided for simplicity by \({{\left| {{{\mathcal{A}}_{0}}} \right|}^{2}}\) can be written as a sum of two terms \(\sum\nolimits_{k = j}^A {\left\langle n \right|{{e}^{{ - i{\mathbf{q}}\mathop {{\mathbf{\hat {X}}}}\nolimits_j }}}{{e}^{{i{\mathbf{q}}\mathop {{\mathbf{\hat {X}}}}\nolimits_k }}}\left| n \right\rangle } \) + \(\sum\nolimits_{j,k \ne j}^A {\left\langle n \right|} {{e}^{{ - i{\mathbf{q}}\mathop {{\mathbf{\hat {X}}}}\nolimits_j }}}{{e}^{{i{\mathbf{q}}\mathop {{\mathbf{\hat {X}}}}\nolimits_k }}}\left| n \right\rangle \), the first of which is \(\sum\nolimits_{k = j}^A 1 \), and the second is \(G({\mathbf{q}})\sum\nolimits_{j,k \ne j}^A 1 \), and this results in \(A + A(A - 1)G({\mathbf{q}})\). Hence follows expression (42).

  9.  Equation (47) can be solved  analytically using the MATHEMATICA package or manually by temporarily designating \(E \equiv \sqrt {{{m}^{2}} + {{p}^{2}} + 2pq + {{q}^{2}}} \) – \(\sqrt {{{m}^{2}} + {{p}^{2}}} \), which yields \({{m}^{2}} + {{p}^{2}} + 2pq + {{q}^{2}}\) = \({{(E + \sqrt {{{m}^{2}} + {{p}^{2}}} )}^{2}}\) = \({{E}^{2}} + 2E\sqrt {{{m}^{2}} + {{p}^{2}}} + {{m}^{2}} + {{p}^{2}}\), and hence, cancelling identical terms on the right and on the left, we have \(2pq + {{q}^{2}}\) = \({{E}^{2}} + 2E\sqrt {{{m}^{2}} + {{p}^{2}}} \)or \(2pq + d = 2E\sqrt {{{m}^{2}} + {{p}^{2}}} \), where the designation \(d = ({{q}^{2}} - {{E}^{2}})\) is introduced. Then, raising to the second power \(4{{E}^{2}}({{m}^{2}} + {{p}^{2}})\) = \({{(2pq + d)}^{2}} = 4{{p}^{2}}{{q}^{2}} + 4dpq + {{d}^{2}}\)and transforming the resulting expression to \(4{{p}^{2}}({{E}^{2}} - {{q}^{2}})\) – \(4dpq + 4{{E}^{2}}{{m}^{2}} - {{d}^{2}} = 0\), we arrive at the quadratic equation \(4d{{p}^{2}} + 4dpq + {{d}^{2}} - 4{{E}^{2}}{{m}^{2}} = 0\). That is, we have the equation \({{p}^{2}} + pq + b = 0\), where \(b = \frac{{{{d}^{2}} - 4{{E}^{2}}{{m}^{2}}}}{{4d}}\), and two solutions \({{p}_{{1,2}}} = - \frac{q}{2}\left( {1 \pm {{{\left[ {\frac{{{{q}^{2}} - 4b}}{{{{q}^{2}}}}} \right]}}^{{1/2}}}} \right)\). Further, \({{q}^{2}} - 4b\) = E2 + \(\frac{{4{{E}^{2}}{{m}^{2}}}}{{{{q}^{2}} - {{E}^{2}}}} = \) \({{q}^{2}}\beta \left( {1 + \frac{{4{{m}^{2}}}}{{{{q}^{2}}}}\frac{1}{{1 - \beta }}} \right)\), where \(\beta = \frac{{{{E}^{2}}}}{{{{q}^{2}}}} \equiv \frac{1}{{{{q}^{2}}}}{{\left( {\frac{{{{{\mathbf{q}}}^{2}}}}{{2{{m}_{A}}}} + \Delta {{\varepsilon }_{{mn}}}} \right)}^{2}}\). Finally, one physical solution has the form \(p = - \frac{q}{2}\left( {1 - \sqrt \beta \sqrt {1 + \frac{{4{{m}^{2}}}}{{{{q}^{2}}}}\frac{1}{{1 - \beta }}} } \right)\).

  10.  Indeed, \({{T}_{A}} = {{E}_{\nu }}\) – \(\frac{{{{{{\Delta }^{2}}{{\varepsilon }_{{mn}}}} \mathord{\left/ {\vphantom {{{{\Delta }^{2}}{{\varepsilon }_{{mn}}}} 2}} \right. \kern-0em} 2} + {{E}_{\nu }}{{m}_{A}} - \Delta {{\varepsilon }_{{mn}}}{{m}_{A}} - \Delta {{\varepsilon }_{{mn}}}{{E}_{\nu }}}}{{{{m}_{A}} + {{E}_{\nu }} - {{E}_{\nu }}\cos\theta - \Delta {{\varepsilon }_{{mn}}}}} - \) \(\Delta {{\varepsilon }_{{mn}}}\). By reducing to the common denominator, removing the parentheses and cancelling out the identical terms, we obtain \({{T}_{A}} = \frac{{E_{\nu }^{2} - E_{\nu }^{2}\cos\theta - {{E}_{\nu }}\Delta {{\varepsilon }_{{mn}}} + \Delta {{\varepsilon }_{{mn}}}{{E}_{\nu }}\cos\theta + {{{{\Delta }^{2}}{{\varepsilon }_{{mn}}}} \mathord{\left/ {\vphantom {{{{\Delta }^{2}}{{\varepsilon }_{{mn}}}} 2}} \right. \kern-0em} 2}}}{{{{m}_{A}} + {{E}_{\nu }} - {{E}_{\nu }}\cos\theta - \Delta {{\varepsilon }_{{mn}}}}}\) = \(\frac{{{{E}_{\nu }}({{E}_{\nu }} - \Delta {{\varepsilon }_{{mn}}})(1 - \cos\theta ) + {{{{\Delta }^{2}}{{\varepsilon }_{{mn}}}} \mathord{\left/ {\vphantom {{{{\Delta }^{2}}{{\varepsilon }_{{mn}}}} 2}} \right. \kern-0em} 2}}}{{{{m}_{A}} + {{E}_{\nu }}(1 - \cos\theta ) - \Delta {{\varepsilon }_{{mn}}}}}.\) Hence, considering \({{m}_{A}} \gg {{E}_{\nu }},{{m}_{A}} \gg \Delta {{\varepsilon }_{{mn}}}\), there follows formula (58).

  11.  Derivation of expression (73) for the hadronic current \(h_{{mn}}^{\mu }\) from (70) is given in appendix 10.2.1.

  12.  Actually, \(i{{\mathcal{M}}_{{mn}}} = i\frac{{{{G}_{{\text{F}}}}}}{{\sqrt 2 }}{{l}_{\mu }}(k,k{\kern 1pt} ')H_{{mn}}^{\mu }({{P}_{n}},P_{m}^{'}) = i\frac{{{{G}_{{\text{F}}}}}}{{\sqrt 2 }}{{l}_{\mu }}(k,k{\kern 1pt} ')\) × \(\sqrt {2P_{n}^{0}2P{\kern 1pt}_{m}^{'0}} h_{{mn}}^{\mu }\) \( = i\frac{{{{G}_{{\text{F}}}}}}{{\sqrt 2 }}{{l}_{\mu }}(k,k{\kern 1pt} ')\sqrt {\frac{{P_{n}^{0}P{\kern 1pt}_{m}^{'0}}}{{{{E}_{{{\mathbf{\bar {p}}}}}}{{E}_{{{\mathbf{\bar {p}}} + {\mathbf{q}}}}}}}} \sum\limits_{k = 1}^A {\left\langle {m\left| {{{e}^{{i{\mathbf{q}}\mathop {{\mathbf{\hat {X}}}}\nolimits_k }}}} \right|n} \right\rangle } \,\) × \(\chi _{m}^{ * }(\{ {{r}^{{(k)}}}\} ){{\chi }_{n}}(\{ r\} )\bar {u}({\mathbf{\bar {p}}} + {\mathbf{q}},{{s}_{k}})O_{k}^{\mu }u({\mathbf{\bar {p}}},{{r}_{k}}).\)

  13.  The square of the matrix element \({{\left| {i{{\mathcal{M}}_{{mn}}}} \right|}^{2}}\) does not depend on the azimuth angle ϕ (see appendix 10.4 or formulas from Section 6.3), and therefore the integration was carried out over it.

  14.  For details see section 10.3.

  15.  This approximation can be a subject of special consideration in the future.

  16.  In the explicit form: \(\begin{gathered} \frac{{d\sigma _{{{\text{inc}}}}^{\nu }}}{{d{{T}_{A}}}} = \frac{{2G_{{\text{F}}}^{2}{{m}_{A}}}}{\pi }\sum\limits_{f = p,n} {{{F}_{f}}} \left\{ {{{A}^{f}}\left[ {{{{(g_{L}^{f})}}^{2}} + {{{(g_{R}^{f})}}^{2}}{{{(1 - y)}}^{2}} - g_{L}^{f}g_{R}^{f}\frac{{2{{m}^{2}}y}}{{s - {{m}^{2}}}}} \right]} \right. + ( + \Delta {{A}^{f}})\left[ {g_{L}^{f} - g_{R}^{f}(1 - y)} \right]\left. {\left[ {g_{L}^{f} + g_{R}^{f}\left( {1 - y\frac{{s + {{m}^{2}}}}{{s - {{m}^{2}}}}} \right)} \right]} \right\}, \\ \frac{{d\sigma _{{{\text{inc}}}}^{{\bar {\nu }}}}}{{d{{T}_{A}}}} = \frac{{2G_{{\text{F}}}^{2}{{m}_{A}}}}{\pi }\sum\limits_{f = p,n} {{{F}_{f}}} \left\{ {{{A}^{f}}\left[ {{{{(g_{R}^{f})}}^{2}} + {{{(g_{L}^{f})}}^{2}}{{{(1 - y)}}^{2}} - g_{L}^{f}g_{R}^{f}\frac{{2{{m}^{2}}y}}{{s - {{m}^{2}}}}} \right]} \right. + \left. {( - \Delta {{A}^{f}})\left[ {g_{R}^{f} - g_{L}^{f}(1 - y)} \right]\left[ {g_{R}^{f} + g_{L}^{f}\left( {1 - y\frac{{s + {{m}^{2}}}}{{s - {{m}^{2}}}}} \right)} \right]} \right\}. \\ \end{gathered} \)

  17.  More details about these form factors are given in appendix 8.1.2.

  18.  With few exclusions, e.g., the DAMA/LIBRA collaboration [91, 92].

  19.  In the COHERENT experiment, target nuclei have a nonzero spin, and therefore, according to formulas (128) and (129), antineutrino-induced CEvNS differs from neutrino-induced CEvNS.

  20.  Since \({{E}_{{\mathbf{p}}}}{{\delta }^{3}}({\mathbf{p}} - {\mathbf{k}}) = {{E}_{{{\mathbf{p}}*}}}{{\delta }^{3}}({\mathbf{p}}{\text{*}} - \,\,{\mathbf{k}}*)\).

  21.  To verify the transformation, one can substitute the upper expression (177) into the lower one, take into account the spinor normalization \({{u}^{\dag }}({\mathbf{p}},s)u({\mathbf{p}},s) = 2{{E}_{{\mathbf{p}}}}\), that \(\int {\frac{{d{\mathbf{x}}{{e}^{{i({\mathbf{p}}{\kern 1pt} '\,\, - {\mathbf{p}}){\mathbf{x}}}}}}}{{{{{(2\pi )}}^{3}}}}} = \) \({{\delta }^{3}}({\mathbf{p}}{\kern 1pt} '\,\, - {\mathbf{p}})\), and obtain the identity

    $$\begin{gathered} \tilde {\psi }({\mathbf{p}},s) = \int {d{\mathbf{x}}\frac{{{{u}^{\dag }}({\mathbf{p}},s){{e}^{{ - i{\mathbf{px}}}}}}}{{\sqrt {2{{E}_{{\mathbf{p}}}}} }}} \int {\frac{{d{\mathbf{p}}{\kern 1pt} '}}{{{{{(2\pi )}}^{3}}}}} u({\mathbf{p}}{\kern 1pt} ',s) \\ \times \,\,\frac{{{{e}^{{i{\mathbf{p}}{\kern 1pt} '{\mathbf{x}}}}}}}{{\sqrt {2{{E}_{{{\mathbf{p'}}}}}} }}\tilde {\psi }({\mathbf{p}}{\kern 1pt} ',s) = \int {d{\mathbf{p}}{\kern 1pt} '} \int {\frac{{d{\mathbf{x}}{{e}^{{i({\mathbf{p}}{\kern 1pt} '\,\, - {\mathbf{p}}){\mathbf{x}}}}}}}{{{{{(2\pi )}}^{3}}}}} \\ \times \,\,\frac{{{{u}^{\dag }}({\mathbf{p}},s)u({\mathbf{p}}{\kern 1pt} ',s)}}{{\sqrt {2{{E}_{{\mathbf{p}}}}} \sqrt {2{{E}_{{{\mathbf{p'}}}}}} }}\tilde {\psi }({\mathbf{p}}{\kern 1pt} ',s) \\ = \frac{{{{u}^{\dag }}({\mathbf{p}},s)u({\mathbf{p}},s)}}{{2{{E}_{{\mathbf{p}}}}}}\tilde {\psi }({\mathbf{p}},s) = \tilde {\psi }({\mathbf{p}},s). \\ \end{gathered} $$

  22.  Or, considering (174), we calculate the matrix element \(\left\langle {{\mathbf{x}}\left| {{{e}^{{i{\mathbf{q\hat {X}}}}}}} \right|{\mathbf{p}}} \right\rangle = \int {\frac{{d{\mathbf{p}}{\kern 1pt} '}}{{{{{(2\pi )}}^{3}}}}} {{e}^{{i{\mathbf{q\hat {X}}}}}}\frac{{e{\kern 1pt} '}}{{\sqrt {2{{E}_{{{\mathbf{p}}{\kern 1pt} '}}}} }}\left\langle {\left. {{\mathbf{p}}{\kern 1pt} '} \right|{\mathbf{p}}} \right\rangle \). Since \({{e}^{{i{\mathbf{q\hat {X}}}}}} \cdot {{e}^{{i{\mathbf{xp}}}}} = {{e}^{{i{\mathbf{x}}({\mathbf{q}} + {\mathbf{p}})}}}\), then \(\left\langle {{\mathbf{x}}\left| {{{e}^{{i{\mathbf{q\hat {X}}}}}}} \right|{\mathbf{p}}} \right\rangle \int {\frac{{d{\mathbf{p}}{\kern 1pt} '}}{{{{{(2\pi )}}^{3}}}}} \frac{{{{e}^{{i{\mathbf{qx}}}}}{{e}^{{i{\mathbf{xp}}{\kern 1pt} '}}}}}{{\sqrt {2{{E}_{{{\mathbf{p}}{\kern 1pt} '}}}} }}{{(2\pi )}^{3}}2{{E}_{{\mathbf{p}}}}\delta ({\mathbf{p}} - {\mathbf{p}}{\kern 1pt} ')\) = \(\sqrt {2{{E}_{{\mathbf{p}}}}} {{e}^{{i{\mathbf{x}}({\mathbf{q}} + {\mathbf{p}})}}}\). Definition (172) of the scalar product can be written in the form \({{e}^{{i{\mathbf{x}}({\mathbf{q}} + {\mathbf{p}})}}} = \frac{{\left\langle {\left. {\mathbf{x}} \right|{\mathbf{q}} + {\mathbf{p}}} \right\rangle }}{{\sqrt {2{{E}_{{{\mathbf{p}} + {\mathbf{q}}}}}} }}\), from which follows (178).

  23.  This is verified by direct substitution of \(\psi (\{ x\} )\) into \(\tilde {\psi }(\{ p\} )\) and the use of the spinor normalization condition

    $$\begin{gathered} \tilde {\psi }(\{ p\} \equiv ({{p}_{i}},{{s}_{i}},{{m}_{i}})) \\ = \int {\left( {\prod\limits_i^n {d{{{\mathbf{x}}}_{i}}\frac{{u_{{{{m}_{i}}}}^{\dag }({{{\mathbf{p}}}_{i}},{{s}_{i}})}}{{\sqrt {2{{E}_{{{{{\mathbf{p}}}_{i}}}}}} }}{{e}^{{ - i{{{\mathbf{p}}}_{i}}{{{\mathbf{x}}}_{i}}}}}} } \right)} \\ \times \,\,\int {\left( {\prod\limits_j^n {\frac{{d{\mathbf{p}}_{j}^{'}}}{{{{{(2\pi )}}^{3}}\sqrt {2{{E}_{{{\mathbf{p}}_{j}^{'}}}}} }}{{u}_{{{{m}_{j}}}}}({\mathbf{p}}_{j}^{'},{{s}_{j}}){{e}^{{i{\mathbf{p}}_{j}^{'}{{{\mathbf{x}}}_{i}}}}}} } \right)} \tilde {\psi }(\{ p{\kern 1pt} '\} ) \\ = \int {\prod\limits_{i,j}^n {\underbrace {\frac{{d{{{\mathbf{x}}}_{i}}}}{{{{{(2\pi )}}^{3}}}}{{e}^{{i{{{\mathbf{x}}}_{i}}({\mathbf{p}}_{j}^{'} - {{{\mathbf{p}}}_{i}})}}}}_{{{\delta }^{3}}({{{\mathbf{p}}}_{i}} - {\mathbf{p}}_{j}^{'})}} } \frac{{d{{{\mathbf{p}}}_{j}}}}{{\sqrt {2{{E}_{{{{{\mathbf{p}}}_{i}}}}}2{{E}_{{{\mathbf{p}}_{j}^{'}}}}} }} \\ \times \,\,\underbrace {u_{{{{m}_{i}}}}^{\dag }({{{\mathbf{p}}}_{i}},{{s}_{i}}){{u}_{{{{m}_{j}}}}}({\mathbf{p}}_{j}^{'},{{s}_{j}})}_{2{{E}_{{{\mathbf{p}}_{j}^{'}}}}{{\delta }_{{{{s}_{i}},{{s}_{j}}}}}{{\delta }_{{{{m}_{i}},{{m}_{j}}}}}{{\delta }^{3}}({{{\mathbf{p}}}_{i}} - {\mathbf{p}}_{j}^{'})}\tilde {\psi }(\{ p{\kern 1pt} '\} \\ = (p_{j}^{'},s_{j}^{'},m_{j}^{'})) = \tilde {\psi }({{p}_{i}},{{s}_{i}},{{m}_{i}}). \\ \end{gathered} $$

  24.  Considering (60), from the relation \({{E}_{\nu }} - E_{\nu }^{'} - \Delta {{\varepsilon }_{{mn}}}\) = \(\sqrt {m_{A}^{2} + E_{\nu }^{2} + {{{(E_{\nu }^{'})}}^{2}} - 2E_{\nu }^{'}{{E}_{\nu }}\cos\theta } - {{m}_{A}}\) we have \({{({{E}_{\nu }} - E_{\nu }^{'} - \Delta {{\varepsilon }_{{mn}}} + {{m}_{A}})}^{2}}\) = \(m_{A}^{2} + E_{\nu }^{2} + {{(E_{\nu }^{'})}^{2}} - 2E_{\nu }^{'}{{E}_{\nu }}\cos\theta \), from which, using expansion (229), we obtain \(\Delta \varepsilon _{{mn}}^{2} - 2{{E}_{\nu }}\mathop {E'}\nolimits_\nu - 2{{E}_{\nu }}\Delta {{\varepsilon }_{{mn}}}\) + \(2{{E}_{\nu }}{{m}_{A}} + 2E_{\nu }^{'}\Delta {{\varepsilon }_{{mn}}} - 2E_{\nu }^{'}{{m}_{A}}\) – \(2{{m}_{A}}\Delta {{\varepsilon }_{{mn}}} = - 2E_{\nu }^{'}{{E}_{\nu }}\cos\theta .\)Then from \(\Delta \varepsilon _{{mn}}^{2} - 2{{m}_{A}}\Delta {{\varepsilon }_{{mn}}} + \) \(2{{E}_{\nu }}({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}})\) = \(2E_{\nu }^{'}({{E}_{\nu }} - {{E}_{\nu }}\cos\theta - \Delta {{\varepsilon }_{{mn}}} + {{m}_{A}})\) there follows formula (227).

  25.  From (228) it follows that \({{T}_{A}}(\tilde {E}_{\nu }^{'}) = {{E}_{\nu }} - E_{\nu }^{'} - \Delta {{\varepsilon }_{{mn}}}\), where \(\tilde {E}_{\nu }^{'} = \tfrac{{{{E}_{\nu }}({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}}) + {{\Delta \varepsilon _{{mn}}^{2}} \mathord{\left/ {\vphantom {{\Delta \varepsilon _{{mn}}^{2}} 2}} \right. \kern-0em} 2} - {{m}_{A}}\Delta {{\varepsilon }_{{mn}}}}}{{{{m}_{A}} - \Delta {{\varepsilon }_{{mn}}} + {{E}_{\nu }}(1 - \cos\theta )}}\). That is, \({{T}_{A}}(\tilde {E}_{\nu }^{'}) = {{E}_{\nu }} - \Delta {{\varepsilon }_{{mn}}}\) – \(\frac{{{{E}_{\nu }}({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}}) + {{\Delta \varepsilon _{{mn}}^{2}} \mathord{\left/ {\vphantom {{\Delta \varepsilon _{{mn}}^{2}} 2}} \right. \kern-0em} 2} - {{m}_{A}}\Delta {{\varepsilon }_{{mn}}}}}{{{{m}_{A}} - \Delta {{\varepsilon }_{{mn}}} + {{E}_{\nu }}(1 - \cos\theta )}}\) \( = \,\,\,\frac{{({{E}_{\nu }} - \Delta {{\varepsilon }_{{mn}}})({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}} + {{E}_{\nu }}(1 - \cos\theta ))}}{{{{m}_{A}} - \Delta {{\varepsilon }_{{mn}}} + {{E}_{\nu }}(1 - \cos\theta )}}\) \( - \,\,\,\,\frac{{{{E}_{\nu }}({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}}) + {{\Delta \varepsilon _{{mn}}^{2}} \mathord{\left/ {\vphantom {{\Delta \varepsilon _{{mn}}^{2}} 2}} \right. \kern-0em} 2} - {{m}_{A}}\Delta {{\varepsilon }_{{mn}}}}}{{{{m}_{A}} - \Delta {{\varepsilon }_{{mn}}} + {{E}_{\nu }}(1 - \cos\theta )}}.\) Then, \({{T}_{A}}(\tilde {E}_{\nu }^{'})\left[ {{{m}_{A}} - } \right.\) \(\left. {\Delta {{\varepsilon }_{{mn}}} + {{E}_{\nu }}(1 - \cos\theta )} \right] = ({{E}_{\nu }} - \Delta {{\varepsilon }_{{mn}}})\left[ {({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}})} \right. + {{E}_{\nu }}\) \(\left. {(1 - \cos\theta )} \right]\) – \([{{E}_{\nu }}({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}}) + {{\Delta \varepsilon _{{mn}}^{2}} \mathord{\left/ {\vphantom {{\Delta \varepsilon _{{mn}}^{2}} 2}} \right. \kern-0em} 2}\,\,\,\, - \) \({{m}_{A}}\Delta {{\varepsilon }_{{mn}}}] = \) \(({{E}_{\nu }} - \Delta {{\varepsilon }_{{mn}}})({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}})\) \( + \,\,({{E}_{\nu }} - \Delta {{\varepsilon }_{{mn}}}){{E}_{\nu }}(1 - \cos\theta ) - \) \({{E}_{\nu }}({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}}) - {{\Delta \varepsilon _{{mn}}^{2}} \mathord{\left/ {\vphantom {{\Delta \varepsilon _{{mn}}^{2}} 2}} \right. \kern-0em} 2} + {{m}_{A}}\Delta {{\varepsilon }_{{mn}}} = {{E}_{\nu }}(1 - \cos\theta )\) + \(({{E}_{\nu }} - \Delta {{\varepsilon }_{{mn}}})\) + \({{E}_{\nu }}({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}}) - {{E}_{\nu }}({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}})\) \( - \,\,{{m}_{A}}\Delta {{\varepsilon }_{{mn}}} + \Delta \varepsilon _{{mn}}^{2} - {{\Delta \varepsilon _{{mn}}^{2}} \mathord{\left/ {\vphantom {{\Delta \varepsilon _{{mn}}^{2}} 2}} \right. \kern-0em} 2} + {{m}_{A}}\Delta {{\varepsilon }_{{mn}}} = {{E}_{\nu }}(1 - \cos\theta )\) × \(({{E}_{\nu }} - \Delta {{\varepsilon }_{{mn}}}) + \Delta \varepsilon _{{mn}}^{2} - {{\Delta \varepsilon _{{mn}}^{2}} \mathord{\left/ {\vphantom {{\Delta \varepsilon _{{mn}}^{2}} 2}} \right. \kern-0em} 2} \to \) (230).

  26. Indeed, raising both sides to the second power and using expansion (229), we have

    $$\begin{gathered} \cos\theta = \frac{{m_{A}^{2} + E_{\nu }^{2} + {{{(E_{\nu }^{'})}}^{2}} - {{{({{E}_{\nu }} - E_{\nu }^{'} + {{m}_{A}} - \Delta {{\varepsilon }_{{mn}}})}}^{2}}}}{{2E_{\nu }^{'}{{E}_{\nu }}}} \\ = \frac{{ - \Delta \varepsilon _{{mn}}^{2} + 2{{E}_{\nu }}E_{\nu }^{'} - 2{{E}_{\nu }}{{m}_{A}} + 2{{E}_{\nu }}\Delta {{\varepsilon }_{{mn}}} + 2E_{\nu }^{'}{{m}_{A}} - 2E_{\nu }^{'}\Delta {{\varepsilon }_{{mn}}} + 2{{m}_{A}}\Delta {{\varepsilon }_{{mn}}}}}{{2E_{\nu }^{'}{{E}_{\nu }}}} \\ = 1 + \frac{{E_{\nu }^{'}({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}}) - {{E}_{\nu }}({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}}) + \Delta {{\varepsilon }_{{mn}}}({{m}_{A}} - {{\Delta {{\varepsilon }_{{mn}}}} \mathord{\left/ {\vphantom {{\Delta {{\varepsilon }_{{mn}}}} 2}} \right. \kern-0em} 2})}}{{\mathop {E'}\nolimits_\nu {{E}_{\nu }}}} \\ = 1 + \frac{{({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}})}}{{{{E}_{\nu }}}} + \frac{{ - {{E}_{\nu }}({{m}_{A}} - \Delta {{\varepsilon }_{{mn}}}) + \Delta {{\varepsilon }_{{mn}}}({{m}_{A}} - {{\Delta {{\varepsilon }_{{mn}}}} \mathord{\left/ {\vphantom {{\Delta {{\varepsilon }_{{mn}}}} 2}} \right. \kern-0em} 2})}}{{\mathop {E'}\nolimits_\nu {{E}_{\nu }}}}. \\ \end{gathered} $$

  27.  Since \({{{\mathbf{n}}}_{{{{k}_{i}}}}} \cdot \sigma = (0,0,1) \cdot \left[ {\left( {\begin{array}{*{20}{c}} 0&1 \\ 1&0 \end{array}} \right),\left( {\begin{array}{*{20}{c}} 0&{ - i} \\ i&0 \end{array}} \right),\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&{ - 1} \end{array}} \right)} \right] = \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&{ - 1} \end{array}} \right)\).

  28.  Because \(\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&{ - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right) = - \left( {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right)\) and \(\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&{ - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right) = + \left( {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right).\)

  29.  From here on the following trigonometric relations are used: \({{e}^{{i\phi }}} = \cos\phi + i\sin\phi \to - i\cos\varphi + \sin\phi = - i{{e}^{{i\phi }}},\) \(i\cos\phi + \sin\varphi = i{{e}^{{ - i\phi }}}.\) \(cos\theta = \sin\theta = 2\sin\frac{\theta }{2}\cos\frac{\theta }{2},\) \(\tan\frac{\theta }{2}\sin\theta + \cos\theta = 1,\) \(\cot\frac{\theta }{2}\sin\theta - \cos\theta = 1.\)

  30.  A possibility of a change in (anti)neutrino helicity after the interaction is not considered. Though, from the point of view of searching for manifestations of New Physics, this effect may in principle have a right to exist if, for example, one assumes nonzero (anti)neutrino masses .

  31.  Or using in parallel the MATHEMATICA package.

  32.  The sign (\( - \)) temporarily separated out below reflects the (abovementioned and insignificant) difference in the definition of the nucleon initial spinor \({{\chi }_{ + }}({{{\mathbf{p}}}_{i}})\): \({{\chi }_{ + }}{{({{{\mathbf{p}}}_{i}})}_{{{\text{here}}}}}\) in (246) \( = - {{\chi }_{ + }}({{{\mathbf{p}}}_{i}})\) in (83) from [19].

  33. \({{g}_{{\mu \nu }}} = \left( {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&{ - 1}&0&0 \\ 0&0&{ - 1}&0 \\ 0&0&0&{ - 1} \end{array}} \right)\). For neutrinos, \(({{l}_{\nu }},{{h}_{{r{\kern 1pt} 'r}}})\bar {u}(k{\kern 1pt} ', - 1){{\gamma }^{\mu }}(1 - {{\gamma }^{5}})\) \(u(k, - 1)\bar {u}(p{\kern 1pt} ',r{\kern 1pt} '){{\gamma }_{\mu }}({{g}_{V}} - {{g}_{A}}{{\gamma }^{5}})u(p,r)\).

  34.  By virtue of the explicit equations for the four-spinors \(u({\mathbf{k}})\) and \(v({\mathbf{k}})\), and procedures for deriving formulas (262) and (263).

  35.  In [19] only the neutrino case was considered. Unfortunately, in formula (C33), \((l,h_{{ + + }}^{\eta }) = 8(s - {{m}^{2}})\cos\frac{\theta }{2}\) × \(\left( {{{g}_{L}} - {{g}_{R}}{{\sin}^{2}}\frac{\theta }{2}\frac{m}{{\sqrt s }}(1 - \frac{m}{{\sqrt s }})} \right)\), an unhappy misprint was made (the sign \( - \) of \({{g}_{R}}\), cf. (299)).

  36.  For example, for any structure similar to the current like \(Q_{{ + - }}^{\eta } = \eta _{ + }^{\dag }({{{\mathbf{p}}}_{f}})Q{{\eta }_{ - }}({{{\mathbf{p}}}_{i}})\) we have in the helicity basis, according to transformation matrices (302), \([\left[ {\sin\tfrac{\theta }{2}\chi _{ + }^{\dag }({{{\mathbf{p}}}_{f}}) + {{e}^{{ - i\phi }}}\cos\tfrac{\theta }{2}\chi _{ - }^{\dag }({{{\mathbf{p}}}_{f}})} \right]Q[( \pm 1){{\chi }_{ + }}({{{\mathbf{p}}}_{i}})]\), from where follows \(Q_{{ + - }}^{\eta } = ( \pm 1)\left\{ {\sin\tfrac{\theta }{2}Q_{{ + + }}^{\chi } + {{e}^{{ - i\phi }}}\cos\tfrac{\theta }{2}Q_{{ - + }}^{\chi }} \right\}\), which is equivalent to the top formula in (304).

  37.  In these formulas, \({{\chi }_{ + }}({{{\mathbf{p}}}_{i}}) = - {{\chi }_{ + }}{{({{{\mathbf{p}}}_{i}})}_{{{\text{here}}}}}\) for the incoming nucleon, as in [19].

  38.  In the squares of the scalars the dependence on \({{\chi }_{ + }}({{{\mathbf{p}}}_{i}}) = - {{\chi }_{ + }}{{({{{\mathbf{p}}}_{i}})}_{{{\text{mine}}}}}\) disappears.

  39.  A good approximation, when \(m \equiv {{m}_{p}} \equiv {{m}_{n}}\).

  40.  For convenience, they are given below in terms of a and b, divided by the factor \(64{{(s - {{m}^{2}})}^{2}}\),

    $$\begin{gathered} {{\left| {\left( {{{l}_{\nu }} \cdot h_{{ - - }}^{\eta }} \right)} \right|}^{2}} = (1 - a)g_{R}^{2}{{\left[ {1 - (1 - b)a} \right]}^{2}},\,\,\,\,{{\left| {\left( {{{l}_{{\bar {\nu }}}} \cdot h_{{ - - }}^{\eta }} \right)} \right|}^{2}} = (1 - a){{\left[ {{{g}_{R}} + {{g}_{L}}b(1 - b)a} \right]}^{2}}, \\ {{\left| {\left( {{{l}_{\nu }} \cdot h_{{ + + }}^{\eta }} \right)} \right|}^{2}} = (1 - a)g_{L}^{2}{{\left[ {1 - (1 - b)a} \right]}^{2}},\,\,\,\,{{\left| {\left( {{{l}_{\nu }} \cdot h_{{ + + }}^{\eta }} \right)} \right|}^{2}} = (1 - a){{\left[ {{{g}_{L}} + {{g}_{L}}b(1 - b)a} \right]}^{2}}, \\ {{\left| {\left( {{{l}_{\nu }} \cdot h_{{ - + }}^{\eta }} \right)} \right|}^{2}} = a{{({{g}_{L}} - {{g}_{R}}b\left[ {1 - (1 - b)a} \right])}^{2}},\,\,\,\,{{\left| {\left( {{{l}_{\nu }} \cdot h_{{ + - }}^{\eta }} \right)} \right|}^{2}} = a{{(1 - a)}^{2}}g_{R}^{2}{{(1 - b)}^{2}}, \\ {{\left| {\left( {{{l}_{\nu }} \cdot h_{{ + - }}^{\eta }} \right)} \right|}^{2}} = a{{({{g}_{R}} - {{g}_{L}}b\left[ {1 - (1 - b)a} \right])}^{2}},\,\,\,\,{{\left| {\left( {{{l}_{{\bar {\nu }}}} \cdot h_{{ - + }}^{\eta }} \right)} \right|}^{2}} = a{{(1 - a)}^{2}}g_{L}^{2}{{(1 - b)}^{2}}. \\ \end{gathered} $$

  41. \(X = 1 - 2(1 - b)a + {{(1 - b)}^{2}}{{a}^{2}} + (a - {{a}^{2}}){{(1 - b)}^{2}} = 1 - 2a + 2ab + (a + 2ab + a{{b}^{2}}) = 1 - a + a{{b}^{2}}.\)

  42. \(\begin{gathered} Z(P,Q) = a{{(P - \left[ {Qb - Qb(1 - b)a} \right])}^{2}} + (1 - a){{[P + Qb(1 - b)a]}^{2}} = (1 - a){{[P + Qab(1 - b)]}^{2}} + a{{[[P + Qab(1 - b)] - Qb]}^{2}} \\ = {{[P + Qab(1 - b)]}^{2}}\underbrace { - a{{{[P + Qab(1 - b)]}}^{2}} + a{{{[P + Qab(1 - b)]}}^{2}}}_0 + 2a[ - P - Qab(1 - b)]Qb + a{{Q}^{2}}{{b}^{2}} = {{P}^{2}} + 2PQab(1 - b) \\ + \,\,{{Q}^{2}}{{a}^{2}}{{b}^{2}}{{(1 - b)}^{2}} - 2PQab - 2{{Q}^{2}}{{a}^{2}}{{b}^{2}}(1 - b) + a{{Q}^{2}}{{b}^{2}} = {{P}^{2}} - 2PQa{{b}^{2}} + {{Q}^{2}}{{a}^{2}}{{b}^{2}}(1 - 2b + {{b}^{2}}) - 2{{Q}^{2}}{{a}^{2}}{{b}^{2}}(1 - b) \\ + \,\,a{{Q}^{2}}{{b}^{2}}\underbrace { + 2PQab - 2PQab}_0 = {{P}^{2}} - 2PQa{{b}^{2}} + {{Q}^{2}}{{a}^{2}}{{b}^{2}} + 2{{Q}^{2}}{{a}^{2}}{{b}^{3}} + {{Q}^{2}}{{a}^{2}}{{b}^{2}} + 2{{Q}^{2}}{{a}^{2}}{{b}^{3}} + a{{Q}^{2}}{{b}^{2}} \\ = \underbrace {{{P}^{2}} - 2PQa{{b}^{2}} + {{Q}^{2}}{{a}^{2}}{{b}^{4}} + {{Q}^{2}}{{a}^{2}}{{b}^{2}} - 2{{Q}^{2}}{{a}^{2}}{{b}^{2}}}_{{{{(P - Qa{{b}^{2}})}}^{2}}} + a{{Q}^{2}}{{b}^{2}} + \underbrace {2{{Q}^{2}}{{a}^{2}}{{b}^{3}} - 2{{Q}^{2}}{{a}^{2}}{{b}^{3}}}_0 = {{(P - Qa{{b}^{2}})}^{2}} + {{Q}^{2}}a{{b}^{2}}(1 - a). \\ \end{gathered} \)

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ACKNOWLEDGMENTS

The authors are grateful to Yu. Efremenko, A. Konovalov, V. Rubakov, V. Naumov, E. Yakushev, and other colleagues for important comments and discussions. V.A. Bednyakov dedicates this work to the memory of Yu.V. Gaponov who passed away in 2009 (just 10 years before this review was prepared).

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Bednyakov, V.A., Naumov, D.V. Concept of Coherence in Neutrino and Antineutrino Scattering off Nuclei. Phys. Part. Nuclei 52, 39–154 (2021). https://doi.org/10.1134/S1063779620060039

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